This function produces a diagonal matrix
D, and invertible matrices
P and
Q such that
D = PMQ. Warning: even though this function is called the Smith normal form, it doesn't necessarily satisfy the more stringent condition that the diagonal entries
d1, d2, ..., dn of
D satisfy:
d1|d2|...|dn..
i1 : M = matrix{{1,2,3},{1,34,45},{2213,1123,6543},{0,0,0}}
o1 = | 1 2 3 |
| 1 34 45 |
| 2213 1123 6543 |
| 0 0 0 |
4 3
o1 : Matrix ZZ <--- ZZ
|
i2 : (D,P,Q) = smithNormalForm M
o2 = (| 135654 0 0 |, | 1 33471 -43292 0 |, | 171927 -42421 54868 |)
| 0 1 0 | | 0 1 0 0 | | 93042 -22957 29693 |
| 0 0 1 | | 0 0 1 0 | | -74119 18288 -23654 |
| 0 0 0 | | 0 0 0 1 |
o2 : Sequence
|
i3 : D == P * M * Q
o3 = true
|
i4 : (D,P) = smithNormalForm(M, ChangeMatrix=>{true,false})
o4 = (| 135654 0 0 |, | 1 33471 -43292 0 |)
| 0 1 0 | | 0 1 0 0 |
| 0 0 1 | | 0 0 1 0 |
| 0 0 0 | | 0 0 0 1 |
o4 : Sequence
|
i5 : D = smithNormalForm(M, ChangeMatrix=>{false,false}, KeepZeroes=>true)
o5 = | 135654 0 0 |
| 0 1 0 |
| 0 0 1 |
3 3
o5 : Matrix ZZ <--- ZZ
|
This function is the underlying routine used by minimalPresentation in the case when the ring is ZZ, or a polynomial ring in one variable over a field.
i6 : prune coker M
o6 = cokernel | 135654 |
| 0 |
2
o6 : ZZ-module, quotient of ZZ
|
In the following example, we test the result be checking that the entries of
D1, P1 M Q1 are the same. The degrees associated to these matrices do not match up, so a simple test of equality would return false.
i7 : S = ZZ/101[t]
o7 = S
o7 : PolynomialRing
|
i8 : D = diagonalMatrix{t^2+1, (t^2+1)^2, (t^2+1)^3, (t^2+1)^5}
o8 = | t2+1 0 0 0 |
| 0 t4+2t2+1 0 0 |
| 0 0 t6+3t4+3t2+1 0 |
| 0 0 0 t10+5t8+10t6+10t4+5t2+1 |
4 4
o8 : Matrix S <--- S
|
i9 : P = random(S^4, S^4)
o9 = | 43 -50 -41 -35 |
| 23 -40 29 50 |
| 18 12 -12 -20 |
| -13 -9 16 -26 |
4 4
o9 : Matrix S <--- S
|
i10 : Q = random(S^4, S^4)
o10 = | -12 36 48 13 |
| -20 -11 -29 26 |
| -46 19 44 -29 |
| -37 2 -32 48 |
4 4
o10 : Matrix S <--- S
|
i11 : M = P*D*Q
o11 = | -18t10+11t8-11t6+14t4-18t2+29 31t10-47t8+36t6+38t4-39t2+37
| -32t10+42t8-38t6+13t4-10t2-34 -t10-5t8+36t6-38t4+23t2
| 33t10-37t8-27t6+29t4+14t2+28 -40t10+2t8-22t6-4t4+5t2+46
| -48t10-38t8-4t6+17t4-13t2-44 49t10+43t8-14t6-14t4-22t2-16
-----------------------------------------------------------------------
9t10+45t8+3t6-34t4+t2+2 37t10-17t8+44t6+11t4-6t2-20 |
16t10-21t8+22t6-3t4-41t2+21 -24t10-19t8+30t6+35t4+20t2+10 |
34t10-32t8+14t6+24t4-34t2+22 50t10+48t8+40t6+38t4+31t2+35 |
24t10+19t8+35t6-13t4+9t2-39 -36t10+22t8-16t6+34t4+13t2+6 |
4 4
o11 : Matrix S <--- S
|
i12 : (D1,P1,Q1) = smithNormalForm M;
|
i13 : D1 - P1*M*Q1 == 0
o13 = true
|
i14 : prune coker M
o14 = cokernel | t6+3t4+3t2+1 0 0 |
| 0 t4+2t2+1 0 |
| 0 0 t2+1 |
| 0 0 0 |
4
o14 : S-module, quotient of S
|
This routine is under development. The main idea is to compute a Gröbner basis, transpose the generators, and repeat, until we encounter a matrix whose transpose is already a Gröbner basis. This may depend heavily on the monomial order.